Electron Spin Dephasing due to Hyperfine Interactions with a Nuclear Spin Bath

We investigate pure dephasing decoherence (free induction decay and spin echo) of a spin qubit interacting with a nuclear spin bath. While for infinite magnetic field $B$ the only decoherence mechanism is spectral diffusion due to dipolar flip-flops of nuclear spins, with decreasing $B$ the hyperfine-mediated interactions between the nuclear spins become important. We give a theory of decoherence due to these interactions which takes advantage of their long-range nature. For a thermal uncorrelated bath we show that our theory is applicable down to $B\sim 10\mathrm{mT}$, allowing for comparison with recent experiments in GaAs quantum dots.

Figure 1

$W(t)$ due to hf-mediated interactions for FID calculated for two GaAs dots of different sizes and various $B$. The solid lines are the results of the full $\mathbf{T}$-matrix calculations (with convergence achieved at matrix size $M\approx 100$), dashed lines are obtained with $3\times 3$ $\mathbf{T}$ matrix taking into account heteronuclear interactions, the dotted lines use Eq. (7). For GaAs we have used $n_{\alpha }=0.604,0.396,1$, ${\mathcal{A}}_{\alpha }=5.47$, 6.99, $6.53\times {10}^{10}{\mathrm{s}}^{-1}$, and ${\omega }_{\alpha }/B=-6.42$, $-8.16$, $-4.58\times {10}^7{\mathrm{s}}^{-1}{T}^{-1}$ for nuclei of ${}^{69}\mathrm{Ga}$, ${}^{71}\mathrm{Ga}$, and ${}^{75}\mathrm{As}$, respectively. The electron spin splitting is $\Omega =0.88g_{\mathrm{eff}}B[T]\times {10}^{11}{\mathrm{s}}^{-1}$ with $g_{\mathrm{eff}}=-0.44$ in a large quantum dot. At high $B$, the actual decay will be bounded by $\sim 10\mu \mathrm{s}$ due to spectral diffusion [5].

Figure 2

Spin echo decoherence $W^{\mathrm{SE}}(t)$ in GaAs due to heteronuclear processes. The dots are obtained in the $\mathcal{A}/N\ll {\omega }_{\alpha \beta }$, $1/t$ limit, when $W(t)=[1+\frac{1}{2}{R}_2(t){]}^{-1}$, while the solid lines are the results of the calculation with the full $\mathbf{T}$ matrix. The differences between the two approaches on a microsecond time scale are visible for the smaller dot (upper panel), but are negligible for the larger one (lower panel). The decay time due to spectral diffusion (calculated as in Ref. 4) is $>10\mu \mathrm{s}$ in both cases. The GaAs parameters used are the same as in Fig. 1.